# rq.process.object

##### Linear Quantile Regression Process Object

These are objects of class `rq.process.`

They represent the fit of a linear conditional quantile function model.

- Keywords
- regression

##### Details

These arrays are computed by parametric linear programming methods using using the exterior point (simplex-type) methods of the Koenker--d'Orey algorithm based on Barrodale and Roberts median regression algorithm.

##### Generation

This class of objects is returned from the `rq`

function
to represent a fitted linear quantile regression model.

##### Methods

The `"rq.process"`

class of objects has
methods for the following generic
functions:
`effects`

, `formula`

, `labels`

, `model.frame`

, `model.matrix`

, `plot`

, `predict`

, `print`

, `print.summary`

, `summary`

##### Structure

The following components must be included in a legitimate `rq.process`

object.

`sol`

The primal solution array. This is a (p+3) by J matrix whose first row contains the 'breakpoints' \(tau_1, tau_2, \dots, tau_J\), of the quantile function, i.e. the values in [0,1] at which the solution changes, row two contains the corresponding quantiles evaluated at the mean design point, i.e. the inner product of xbar and \(b(tau_i)\), the third row contains the value of the objective function evaluated at the corresponding \(tau_j\), and the last p rows of the matrix give \(b(tau_i)\). The solution \(b(tau_i)\) prevails from \(tau_i\) to \(tau_i+1\). Portnoy (1991) shows that \(J=O_p(n \log n)\).

`dsol`

The dual solution array. This is a n by J matrix containing the dual solution corresponding to sol, the ij-th entry is 1 if \(y_i > x_i b(tau_j)\), is 0 if \(y_i < x_i b(tau_j)\), and is between 0 and 1 otherwise, i.e. if the residual is zero. See Gutenbrunner and Jureckova(1991) for a detailed discussion of the statistical interpretation of dsol. The use of dsol in inference is described in Gutenbrunner, Jureckova, Koenker, and Portnoy (1994).

##### References

[1] Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles,
*Econometrica*, **46**, 33--50.

[2] Koenker, R. W. and d'Orey (1987, 1994).
Computing Regression Quantiles.
*Applied Statistics*, **36**, 383--393, and **43**, 410--414.

[3] Gutenbrunner, C. Jureckova, J. (1991).
Regression quantile and regression rank score process in the
linear model and derived statistics, *Annals of Statistics*,
**20**, 305--330.

[4] Gutenbrunner, C., Jureckova, J., Koenker, R. and
Portnoy, S. (1994) Tests of linear hypotheses based on regression
rank scores. *Journal of Nonparametric Statistics*,
(2), 307--331.

[5] Portnoy, S. (1991). Asymptotic behavior of the number of regression
quantile breakpoints, *SIAM Journal of Scientific
and Statistical Computing*, **12**, 867--883.

##### See Also

`rq`

.

*Documentation reproduced from package quantreg, version 5.54, License: GPL (>= 2)*